EVALUATION OF GOVERNMENT INVESTMENT USING NESTED PROBABILISTIC LINGUISTIC PREFERENCE RELATIONS BASED ON GRAPH THEORY

. Government investment, as a major government function, is closely related to national development and economic growth. It plays a key role to maximize the benefits of this fund, which requires the government to choose the optimal investment plan. Considering the complex and uncertain decision-making environment, we propose the nested probabilistic linguistic preference relation (NPLPR) based on the nested probabilistic linguistic term sets (NPLTSs), to express preference information from the qualitative and quantitative angle. According to graph theory, we define a consistency index and an acceptable consistency of NPLPR to measure the additive consistency. Based on which, we establish a novel algorithm for unacceptable consistent NPLPR to meet the acceptable consistency. Finally, projects in government investment are evaluated by the proposed decision-making method, and some comparative analyses, discussions, and implications are provided from three angles. This study provides a new perspective for scholars to make scientific and rational decisions with the help of technological and economic development in various fields.


Introduction
Government investment could make up for market failure, coordinate the proportion of major investment in the whole society, and then promote economic development and structural optimization.Therefore, it is a necessary means of national macro-economic regulation and control, and plays an important macro-guiding role in social investment and resource allocation.With the development of the society, evaluation information is increasingly uncertain and complex in decision-making problems (Dahooie et al., 2020).Decision makers (DMs) are often unable to take crisp numbers to express their evaluation information because of lacking professional knowledge and complex decision-making environment.Considering that human's cognition is of inevitable fuzziness in practice, people prefer to use linguistic expressions to describe assessment results (Zadeh, 1975), and thus linguistic variables could express the DMs' preferences effectively.At present, many researchers have paid attention to establishing various flexible linguistic models or complex expressions to better describe uncertain information.(Ban et al., 2020).
Since the research of Zadeh in 1975, the theoretical developments of linguistic representation models have continued until now.The first phase is the single linguistic term to express opinion.Later, linguistic models were improved to add probability to the linguistic term considering the uncertainty, such as virtual linguistic terms (VLT) (Xu, 2004), and 2-tuple fuzzy linguistic representation model (2TFLRM) (Herrera & Martinez, 2000).Wang and Hao (2006) proposed a novel 2-tuple fuzzy linguistic representation model based on the concept of symbolic proportion.Alcala et al. (2007) proposed a genetic learning of compact fuzzy rule.In addition, the OWA-based consensus operator (Dong et al., 2010) were further defined to calculation.Due to that the single linguistic term is unable to describe people's views comprehensively, and they often tend to hesitate between different linguistic terms, like "between good and very good".Hesitant Fuzzy Linguistic Term Set (HFLTS) was proposed to deal with describing complex information (Rodríguez et al., 2012).Probabilistic Linguistic Term Set (PLTS) (Pang, et al., 2016), and distributed linguistic representation (Zhang et al., 2014) were proposed to express comprehensive information combined probabilistic information and linguistic term.In the face of increasingly uncertain linguistic information, one-layer linguistic model cannot be expressed effectively.Later, Double Hierarchy Hesitant Fuzzy Linguistic Term Set (DHHFLTS) (Gou et al., 2017), 2-Dimension Linguistic Term Set (2DLTS) (Zhao et al., 2019) were proposed to split linguistic expression.Up to now, HFLTSs and PLTSs have been popular and applied in various fields.HFLTSs focus on collecting possible linguistic terms where DMs have some hesitations on them (Torra, 2010;Rodríguez et al., 2013).In the hesitant fuzzy linguistic environment, recent studies have been focused on the distance and similarity measures (Liao et al., 2014), consistency and consensus (Wu & Xu, 2016), correlation coefficients (Liao et al., 2015b), and decision-making methods (Liao et al., 2015a).Considering different importance degrees of linguistic terms, PLTSs is more reasonable to represent people's preference.Under such an environment, Bai et al. (2017) established a comparison method and developed a rational way to apply in the decisionmaking field.
Under the complex decision-making environment, either one-dimension information or evaluation without degree cannot meet the basic needs to expression for complex decisionmaking problem, such as "very good (60% degree) in the first-class hospital (20% degree)" or "a little bit (20% degree) for good (70% degree)".To measure such the multi-dimensional and uncertainty of linguistic evaluations, Wang et al. (2019a) proposed the Nested Probabilistic-Numerical Linguistic Term Set (NPNLTS), and extended it to a general concept, defined as the nested probabilistic linguistic term set (NPLTS) (Wang et al., 2021).A NPLTS consisted of "outer and inner" structure, could express four situations in terms of the linguistic type.
(1) Case 1: Ordinal variable and ordinal variable.This is suitable to describe complex performance information.For instance, when a doctor asks for a patient about his blood pressure, he may say "a little bit high".(2) Case 2: Ordinal variable and nominal variable.This is applied to express the linguistic information from the overall performance to the local characteristic.For example, when a consumer buys a car, he may first consider the "good" company and then consider the "after-sale service".(3) Case 3: Nominal variable and ordinal variable.This is suitable to describe the linguistic information from the local characteristic to its performance.For example, when a user selects a carrier, he may consider "high bandwidth".(4) Case 4: Nominal variable and nominal variable.This is applied to describe double linguistic information.For instance, when a consumer chooses a mobile phone, he may consider "iPhone and then its price".In this way, multi-dimensional and uncertain information could be expressed with clear structure accurately and comprehensively.To compare the differences of NPLTS and other linguistic models, Table 1 lists the characteristics with some linguistic representation models.The NPLTSs remain the advantages of the PLTSs, and it is necessary and worthful to study NPLTSs in both theory and practice.Up to now, a series of distance and similarity measures of NPLTSs have been established in different cases (Wang et al., 2019b).The nested probabilistic linguistic information has been used to handle the allocation problem of water resources, the maneuvering target tracking, and the consensus-based track association problem (Wang et al., 2020).
DMs prefer to express their judgments through attributes or pairwise comparisons of the alternatives.Preference relation is an efficient and common tool to describe uncertain information over a set of alternatives (Chuang et al., 2020).Currently, there are three preference types to compare two alternatives.The first one is that DM either prefers one to the other.The second one is that DM is indifferent between two alternatives, and the third one is that DM is unable to compare them.Herrera-Viedma et al. (2004) proposed two mathematical models based on the concept of preference relations.These studies have been indicated that it is more accurate for pairwise comparison methods than non-pairwise methods compared with various preference methods.The reason may be that pairwise comparison focuses on the evaluated two alternatives without other alternatives (Meng et al., 2019).Hence, preference relations have become powerful techniques because of describing uncertain information conformed to people's cognitive (Wu & Xu, 2016).It is worthy and necessary to study linguistic preference relations, and it is an important branch to research the linguistic preference relations (Mi et al., 2020).This study mainly focuses on the nested probabilistic linguistic preference relations (NPLPRs), a new preference relation form, and then apply them to the decision-making problems in government investment.Due to that an inconsistent preference relation would result in misleading results, consistency is a key property no matter what preference relations, and it is necessary for consistency checking to guarantee DMs to make logical results.It is a hot topic to study the consistency and consistency checking based on linguistic models, and many scholars have studied them (Yang et al., 2021).Since NPLTSs can deal with uncertain and complex information effectively in decision-making, it is worth to define the consistency of NPLPR to make a leading and rational result.Additionally, transitivity is the basic concept for the consistency (Wang, 1997), because the requirement to characterize consistency is to extend the classical requirements of binary preference under an uncertain environment.Many of the properties, such as minimum transitivity, and additive transitivity (Herrera-Viedma et al., 2004), have been proposed for different preference relations to apply transitivity.Considering that it is suitable to use the additive consistency for fuzzy preference relations (Zhang et al., 2018), and graph theory is a very useful method to describe the consistency of preference relation (Boffey, 1982), this study measures the additive consistency of NPLPR with the help of graph theory.
The contributions of this paper lie in the following aspects: (1) From a new perspective of linguistic expression model, i.e., NPLTSs, the concept of the NPLPR is firstly proposed to better present preference information comprehensively in government investment by pairwise comparison of alternatives.(2) To ensure the rationally and scientifically, an additive consistency of the NPLPR is proposed based on graph theory.(3) To improve the unacceptable consistent NPLPR until it is acceptable and help DMs make the reasonable and effective decisions, two novel algorithms are established.One is an automatic improving algorithm, and another is a decision-making algorithm with the consistent NPLPR.(4) An experimental study concerning the evaluation of investment plans for government is given.Some comparative analyses are conducted from three perspectives including the impact for using NPLPR without checking consistency, various preference relations and the changed adjusted parameter.
The organization of this study is as follows: Section 1 reviews related contents of NPLTSs.Section 2 establishes the research method.In Section 3, we give a case study to show the proposed method step by step, and conduct comparati,ve analyses and discussions.In the last Section, we end the study with conclusions.

Linguistic scale and nested probabilistic linguistic term set
Some concepts are reviewed, including the additive linguistic evaluation scale, NPLTSs and the normalization of NPLTSs.In general, additive linguistic evaluation scale has two types, i.e., the traditional additive linguistic scale (Xu, 2012) and the subscript-symmetric linguistic scale (Xu, 2005).Specifically, the form of traditional additive linguistic scale is , where s a represents a possible value for a linguistic label.And the form of subscript-symmetric linguistic evaluation scale is where s 0 means an indifference, while other labels are distributed around it symmetrically.In this study, subscript-symmetric linguistic evaluation scale is used, and there are some operational laws: (1) if a > b, then s a > s b ; (2) the negation operator is defined as: ( ) a −a = neg s s , and ( ) = 0 0 neg s s .With the linguistic scale, the NPLTSs was proposed as follows (Wang et al., 2021): A nested linguistic term set (NLTS) consists of outer linguistic term set (OLTS) and inner linguistic term set (ILTS) denoted as , , 1,0,1, , I S n   , respectively.The NLTS can be written as s n names as the nested linguistic term.NPLTS is a mapping function from a finite set X to a subset of NLTS S N , denoted as: ) The normalization method is developed as follows (Wang et al., 2021): Let ( ) where ( ) ( ) To compare the comprehensive preference values of alternatives directly, Wang et al.

Consistency of NPLPR
Given a set of alternatives , and the DMs express their preference information through comparing each pair of alternatives using NPLTSs.The NPLPR is defined as follows: A NPLPR The normalized NPLPR (N-NPLPR) Firstly, the consistency of NPLPR consisting of NPLTSs is discussed with one element both in the OLTS and in the ILTS.Specifically, as for any NPLTS, , there are one outer linguistic term with its corresponding probability and one inner linguistic term with its corresponding value, then we call it a special NPLPR denoted as , it holds full evaluation information; when , there are partial assessment information.In such a circumstance, the special preference relation is defined as: Let , and the evaluation information are obtained by The preference information is expressed as special NPLTSs.Given a special NPLPR where , the special N-NPLPR with one element is denoted as Next, we define the special NPLPR satisfied additive consistency.Let , and Motivated by the preference relation graph (P-graph) and the symmetric preference relation graph (S-P-graph), we define the P-graph and the S-P-graph with respect to special NPLPR are defined as follows: Let V A is a weighted P-graph with priority, where are the set of vertices and the set of arcs, respectively.The ( ) is a set of vertices and is a set of arcs.
The ( ) , i j v v from the v i to vertex v j is a directed line segment.The weight and the priority of the arc ( ) w v v and ( ) , To represent the information of a path from the one vertex to another vertex in Pgraph or S-P-graph, the length and the importance of the path are defined as follows: Let ( ) = , G V A be the weighted digraph with priority, and The length of path is the sum of weights of its arcs, denoted as , and the importance of the path is the sum of the priorities of its arcs, denoted as The corresponding special N-NPLPR is:


The P-graph and S-P-graph of B are presented in Figure 1.-2(-1) a) The P-graph b) The S-P-graph Next, we study the situation where at least one NPLTS has multiple elements in a NPLPR.The additively consistent NPLPR is defined as follows: be a NPLPR, where be a NPLPR and V A is a weighted P-graph with priority, where are a set of vertices and a set of arcs, respectively.( ) be a NPLPR and be the corresponding N-NPLPR, a weighted S-P-graph with priority is denoted as are a set of vertices and a set of arcs.The ( ) p k v v are the k-th weight and priority of the ( ) be a NPLPR and In general, we denote as a NPLPR and Figure 2 shows the P-graph and the P-graph with additively consistent of N S P . Figure a) The P-graph of P S N b) The P-graph with additively consistent P S N -

Consistency improving for NPLPR
Consistency is an important issue for any preference relation before decision-making.In this section, a consistency checking and improving algorithm are established, and a NPLPR-based decision-making method is further developed.

Consistency index of NPLPR
A distance measure between two NPLTSs is defined as: Let S S m m P p be the corresponding N-NPLTSs, respectively, which satisfy It satisfies the fundamental properties of distance, i.e., non-negativity, symmetry, and boundedness.Therefore, it is a normalized distance.In the following, we define the distance between any two NPLPRs.Let be the corresponding N-NPLPRs, respectively.Then Furthermore, the consistency index of NPLPR N S P  is denoted as: The smaller the value of CI, the higher the consistency level of N S P  .The values of ( ) N S CI P  were provided for different order n when a = 0.1 and s = 2, shown in Table 2 (Dong et al., 2008).
Based on the discussion above, an automatic improving algorithm for the NPLPR with unacceptable consistency is established, presented as follows: Algorithm 1 , the number of iterations z = 0 and an adjusted parameter ) q ≤ q ≤ 0 1 .
Step Step 5. Let Step 6.Let ( ) , and output the modified NPLPR . Go to Step 7.
When q = 0, the modified NPLPR is the original NPLPR, and when q = 1, the modified NPLPR satisfies additive consistency.The larger the adjusted parameter q, the faster the speed to satisfy consistency.Therefore, Algorithm 1 is convergent.Moreover, the adjusted parameter q is provided by the DMs and depends on specific practical problems.Furthermore, the adjusted parameter q can be regarded as a preference coefficient of the DMs.

Let
be a set of alternatives.The DMs compare the alternatives with a weighting vector n j j j n  , and then a preference relation with NPLTS information can be obtained as follows: is a NPLTS, denoting the preference information that the alternative x i compares to the alternative x j .
Algorithm 2 , the weight vector
Output: The best alternative * i x .
Step Step 5. Improve the NPLPR by Algorithm 1, and obtain the modified . Go to Step 6.
Step 6. Integrate the preference information in with NPLWA by Eq. ( 5).Go to Step 7.
Step 8. Obtain the optimal alternative * i x .Go to Step 9.

A case study
Each project has its advantages and disadvantages, and they are evaluated by experts from an overall perspective.NPLPR, as a new preference relation, is suitable for practical problems of multiple dimension and nested structure.In this section, we consider "new energy development", an important scientific and technological problem to be breakthrough in the world, as an example to explain how the government chooses the overall optimal investment project after expert evaluation and make some comparative analyses to illustrate the reliance of the proposed NPLPR-based method.

Problem description
New energy, such as geothermal energy, ocean energy, biomass energy and nuclear fusion energy, is different from traditional energy, which has just started to be developed and utilized or is being actively studied and to be promoted (Geng et al., 2021).In people's life, the main energy sources used are fossil fuels.With the improvement of social life, it is expected that the world energy consumption will grow at a rate of 2.7% per year in the future.According to the current international common energy forecast, oil resources will be exhausted in 40 years, natural gas resources will be used up in 60 years, coal resources can only be used for 220 years.According to the prediction of the international authority unit, by the 2060s, that is, 2060, the proportion of the global new energy will develop to account for more than 50% of the world's energy composition, become the cornerstone of the future energy of human society, the leading role in the world energy stage.Burning of fossil fuels, is the main factor causing air pollution, and how to use the energy in the development has become a major global problem, and as a result, limit and reduce emissions of carbon dioxide from the burning of various fuels and gases, has become the international community to slow global climate change is an important part of (Wang & Xin, 2020).New energy is a clean energy to protect the ecological environment.Using new energy to gradually reduce and replace the use of fossil energy is a major measure to protect the ecological environment and move towards the road of sustainable economic and social development (Yao et al., 2021).Therefore, it is very important for the government to choose and promote excellent new energy development projects for the development of the country.
In general, the system aims to choose the best investment plan from four alternative plans: -Sustainable capacity.If the invested project can achieve sustained phased results and continue to move forward, it shows that the project can create more new blood for the new energy industry.Of course, the government hopes that the funds invested in the project can play a long-term role and create long-term value for the relevant industries.-Project size.The size of the project scale reflects the importance of the project initiated by the project unit from the side.Larger projects tend to have less risk and contain greater development potential.-Economic benefits.Governments, of course, would like the same amount of money spent to produce a greater degree of financial output.-Environmental friendliness.With the continuous advancement of modernization and the progress of science and technology, environmental protection has brought great challenges.As a national regulator, the government needs to pay attention to environmental protection.It is very important to choose environmentally friendly projects with a low degree of environmental damage.-Talent and technology reserves.The research of new energy is a high-tech research and development project, and the number of cutting-edge talents and cutting-edge technology in the project team has an impact on the actual value of the project results.

Solve the problem
Suppose that four new type projects plans { } 1 2 3 4 , , , x x x x are put forward above which can be invested by government.The system compares each pair of plans using the OLTS and the ILTS as follows:

Comparative analyses and discussions
Some comparative analyses are conducted to show the advantages of the proposed NPLPRbased method by simulation experiments from three perspectives: (1) The impact for NPLPR without checking consistency In this part, we use the NPLPR without checking consistency in the proposed method for the applied above, and compare the result with the proposed method.Table 4 shows the rankings of the projects in the above case study.

Alternative ranking
The proposed method with consistency checking and improving The proposed method without consistency checking and improving In Table 3, the ranking results are a little different by using two methods.x 1 is still the best project.However, the priorities of x 3 and x 4 are reversed.There are two different rankings by using two methods.If the decision-making method based on preference relations without consistency checking and improving, the results may be unreliable.Therefore, it is scientific to obtain the consistent NPLPR to get the optimal alternative through consistency checking and improving process. (

2) The impact for various preference relations
To compare and analyze the results clearly, we use the hesitant fuzzy preference relation (HFPR), the hesitant fuzzy linguistic preference relation (HFLPR) and the probabilistic lin-  The HFPR > > > The HFLPR > > > As we can see, all the best project is x 1 , and the rankings are the same by using NPLPR and PLPR.However, the results are different when using HFPR and HFLPR, the reason may be that they are not considered the preference degrees.Although results are the same based on PLPR and NPLPR, the PLPR cannot reflect the complete preference information than the NPLPR.Therefore, compared with other three preference relation, the ranking results with NPLPR are reasonable and reliable.
(3) The impact for the adjusted parameter θ In the following, we mainly study the impact for the adjusted parameter q in the process of improving consistency.Let m be the number of alternatives, t be the average operation time (AOT).Then, the situation of AOTs with the certain value of q is shown in Figure 3.
AOTs are monotonically increasing with the increase of the m and the decrease of q.Besides, it tends to increase exponentially.Therefore, the adjusted parameter q is an improving preference coefficient from this point of view.In order to see the relationships clearly among the parameters q, m and t, suppose that q is from 0 to 1, m is taken from 3 to 14, and the calculation step size is 0.001.Some simulation experiments are provided after 1000 simulation times, shown in Figure 4.The larger the adjusted parameter q is, the faster the consistency of NPLPR is achieved and the less the AOT is.Similarly, the more the number of the alternative m is, the more the AOT is.As we can see, it is consistent with the actual decision-making process.

Implications
There are various complex decision-making problems in government investment, and the consideration of factors such as national defense, by the government through the fiscal investment or local financial bonds, the use of foreign government grants and the domestic and international financial organizations loan of national finance guarantee way owned or joint venture to build assets investment project (Leeper et al., 2010), to promote the national economy and the development of regional economy, to meet the needs of the social culture and life.The multi-dimensional and uncertain decision-making problem has become increasingly difficult for people to make a scientific and rational decision.The purpose of the following discussions and implications is to provide the meaning of the NPLPR to deal with multi-dimensional and complex practical applications effectively.(1) To describe uncertain evaluation information with nested structure with respect to attributes, and it is suitable to apply for the complex decision-making problems; (2) To express multi-dimensional evaluation information whether the preference degree or feature of alternative, which is realized by ordinal variables and nominal variables; (3) To be applicable to various research directions in terms of the type of elements in the nested linguistic term set, and it could be applied to decision-making, optimization, and discrimination problem.According to discussions and comparative analyses, the proposed NPLPR-based method not only considers the consistency, but also describes quantitative and qualitative information of the linguistic terms, and it is more flexible than other popular preference relations and could better reflect the actual situation in real life.
Results in this study provide a novel decision-making approach with NPLPR, and it benefits researchers, policymakers, and practitioners in several ways.(1) In terms of consistency of NPLPR, scholars could further study other forms, like multiplicative consistency, and then propose advanced decision-making approaches.On the other hand, consistency index is a key factor in preference relations, and how to get the rational and scientific index with various number of experts is also an interesting direction for researchers to make contributions.
(2) NPLPR is a useful tool for policymakers to evaluate projects or resources, because it considers both qualitative and quantitative information, and nested structure, which conforms to decision-making problems in real life.(3) From the perspective of practice, policymakers could not only adopt qualitative or quantitative decision, but also heterogeneous, interactive, and multi-dimensional decision.From the perspective of theory, there are research gap related to different consistency, automatic improved algorithm of inconsistency preference relation, and decision-making approaches, especially under the complex and uncertain environment.These points have important implications for academics, companies, and policymakers.

Conclusions
Government investment is related to the national economy and the life of the public.Due to the limitation of the amount, it is necessary to use appropriate methods to choose the optimal investment projects.Nested probabilistic linguistic term set (NPLTS) is a novel and practical way in information expression from the perspective of its structure, and it is more consistent that the elements of OLTS are ordinal variables, m m P p  meets the requirement that there are all N-NPLTSs in the upper triangular matrix.To analyze the relationship between NPLPR and the NPLTSs in the NPLPR, we first define the following relationship based on the whole score of the NPLTS: Given two NPLTSs v i and v j .( ) are the weight and the priority of the arc ( ) , i j v v , respectively, and ( ) ( )

Figure 1 .
Figure 1.The P-graph and S-P-graph of the special NPLPR B (sources: authors' own research) v are represented the k-th weight and priority of the arc ( ) z + 1. Go back to Step 3.

Figure 3 .Figure 4 .
Figure3.The AOT with various values of q when m is taken from 3 to 14 (sources: authors' own research)

Table 1 .
The characteristics with linguistic representation models (source: our research) Note:The full names of the abbreviations are: 2TFLRM -2-tuple fuzzy linguistic representation model, VLS -Virtual linguistic terms, HFLTS -hesitant fuzzy linguistic term set, PLTS -probabilistic linguistic term set, DHHFLTS -double hierarchy hesitant fuzzy linguistic term set, 2DLTS -2-dimension linguistic term set, NPLTS -nested probabilistic linguistic term set.

Table 2 .
The values of N S P  is supposed to be improved to meet consistent NPLPR ×

Table 4 .
Rankings based on two methods (source: our research)

Table 5 .
Rankings based on four preference relations (source: our research)